Chapter 9: Einstein and Relativity Theory (319 KB) - D Cassidy Books

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3637_CassidyTX_09 6/14/02 12:08 PM Page 422 422 9. EINSTEIN AND RELATIVITY THEORY DERIVATION OF TIME DILATION: THE LIGHT CLOCK The “clock” consists of a stick of length l with a mirror and a photodetector P at each end. A flash of light at one end is reflected by the mirror at the other end and returns to the photodetector next to the light source. Each time a light flash is detected, the clock “ticks” and emits another flash. Diagram (a) below shows the clock as seen by an observer riding with the clock. The observer records the time t between ticks of the clock. For this observer, the total distance traveled by the light pulse during the time t is d 2l. Since the light flash travels at the speed of light c: d 2l ct. So l ct/2. Diagram (b) shows the same clock as seen by an observer who is “stationary” in his or her own framework, with the clock apparatus moving by. This observer observes and records the time t between ticks of the clock. For this observer, the total distance traveled by the light beam is d in time t. Since light travels at the same speed for all observers moving at uniform speed relative to each other, we have dct. Let’s look at the left side of drawing (b). Here the motion of the clock, the vertical distance l, and the motion of the light beam form a right triangle. The base of the triangle is the distance traveled by the clock in time t/2, which is vt/2. The distance the beam travels in reaching the mirror is d/2. Using the Pythagorean theorem, we obtain d 2 l 2 . 2 2 2 From the above, we can substitute dct and l ct/2: ct ct vt vt 2 . 2 2 2 2 2 Mirror R R R R d ′ / 2 d ′ / 2 v Photodetector P P P P (a) (b)
3637_CassidyTX_09 6/14/02 12:08 PM Page 423 9.6 RELATIVITY OF TIME 423 Squaring and canceling like terms, we have c 2 t 2 c 2 t 2 v 2 t 2 . Now, let’s solve for t: c 2 t 2 v 2 t 2 c 2 t 2 , t 2 (c 2 v 2 ) c 2 t 2 , c 2 t 2 t 2 , c 2 v 2 t 2 t 2 , 1 v 2 /c 2 or t t , 1 /c v 2 2 Since 1 v 2 /c 2 is here always less than 1, the denominator is less than 1, and the fraction is larger than t alone. Thus, the time interval t registered by the clock as seen by the stationary observer is “dilated” compared to the time interval t registered by the clock as seen by the observer riding with the clock. In other words, the moving clock appears to run slower as measured by the stationary observer than when the clock is not moving with respect to the observer. Note also the crucial role of Einstein’s second postulate in this derivation. Since d is larger than d, t must be larger than t, in order for the ratios on the right side of both equations to have the same value, c. This means that the time interval (t) for the round trip of the light pulse, as registered on the clock as John observes it, is longer than the time interval (t) registered on the clock as Jane observes it. The surprising conclusion of this thought experiment (which is really a deduction from the postulates of relativity theory) is: Time intervals are not absolute and unchanging, but relative. A clock (such as Jane’s), or any repetitive phenomenon which is moving relative to a stationary observer appears to the stationary observer to run slower than it appears to do when measured by the observer moving with the clock—and it appears to run slower the faster the clock is moving. This is known as time dilation. Just how much slower does a clock seem when it is moving past an observer To get the answer, you can use the diagram in Figure 9.10 of John and Jane and apply the Pythagorean theorem. After a bit of basic algebra (see the derivation in the insert), you obtain the exact relationship between the time elapsed interval registered by a clock that is stationary with respect to the observer (as in the case of Jane)—call it now T s —and the
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Chapter 9: Einstein and Relativity Theory (319 KB) - D Cassidy Books

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